In semiconductor device processing, non-planar surfaces are produced when features, such as electrodes, insulating layers, and other three-dimensional elements, are formed on semiconductor substrates. The resulting non-planar surfaces present difficulties in subsequent processing steps, such as the deposition of wiring stripes, in the manufacture of semiconductor devices. Therefore, it is known in the art to deposit materials on the devices in order to produce a planar or nearly planar surface to improve the yield after subsequent processing.
Among the techniques employed for "planarization" is the application of glasses, such as phosphosilicate glass (PSG), borophosphosilicate glass (BPSG), and spin-on glasses (SOG). These planarization materials are flowable, i.e., tend to flow slowly when raised to a sufficiently high temperature. The flow tends to fill valleys so that a surface of improved planarity is obtained.
In order to obtain a desired surface configuration after the deposition of a planarization material and its reflow, it is helpful to predict the surface configuration based upon the initial surface configuration, the characteristics of the flowable material, and the reflow conditions. An example of a method of predicting a two-dimensional surface topography resulting from the flowing of a flowable material is described by Leon in "Numerical Modeling of Glass Flow and Spin-on Planarization", IEEE Transactions on Computer Aided Design, Volume 7, Number 2, February 1988, pages 168-173, which is incorporated herein by reference. The method described by Leon employs a surface diffusion material flow theory to predict how a two-dimensional surface profile will change as a function of time in an attempt to minimize the surface energy. The diffusion is driven by the chemical potential gradient at the surface of a two-dimensional structure. Whenever the gradient is present and conditions favor diffusion, the flowable material will move, changing the configuration of the surface. The greater the chemical potential gradient, the more rapid the flow for other fixed conditions, e.g., temperature.
Leon's model is described with reference to FIGS. 9-12. In FIG. 9, a BPSG body 1 has a surface 2 and is disposed on a substrate 3. The glass includes a central opening 4 at the surface of the substrate. The BPSG may have been deposited by any known technique followed by etching to prepare the opening 4. In the Leon two-dimensional model, a plurality of points 5 are established at spaced intervals along the surface of the BPSG 1 and the exposed surface of the substrate 3 at the opening 4 in the BPSG 1. Adjacent pairs of points 5 are connected by line segments 6, sometimes referred to as strings. If the temperature of the BPSG 1 is raised higher than about 600.degree. C., the BPSG can flow. In the flow process, treated by Leon as a surface diffusion process, sharp edges tend to become smooth and depressions tend to be filled. Each of the points 5 on the surface of the BPSG 1 can be considered, in combination with its two closest neighbors, to lie along a radius of curvature. An example is one of the rectangular corners shown within the area A of FIG. 9 is shown in greater detail in FIG. 10. The point 5 at the corner has a radius of curvature R. As the BPSG 1 flows and the sharpness of the corner is lost, the radius of curvature of the point 5 initially at the corner increases as a result of the flow, i.e., surface diffusion. Typically, a positive radius of curvature is assigned to points 5 like those at an outside corner, such as is shown in FIG. 10. A negative radius of curvature is assigned to points at inside corners, such as the point of intersection of the substrate 3 in the opening 4 with the BPSG 1 shown in FIG. 9.
According to the Leon method and well known surface diffusion theory, whenever the radius of curvature of a surface is less than infinite, i.e., when the surface is non-planar, there is a gradient in the chemical potential across the curved surface proportional to the surface free energy .gamma., i.e., the surface tension. The total free energy, F, of an area equals .gamma.A where A is the total area and .gamma. is the surface free energy per unit area. The local chemical potential at a particle N on a surface equals the derivative of the surface free energy with respect to the number of particles. Applying these relationships, EQU .mu.=(dF/dN)=.gamma.(dA/dN)=.gamma..OMEGA.((1/R.sub.1)+(1/R.sub.2)).
In the foregoing equation, .OMEGA. is the atomic or molecular volume of the atoms or molecules, i.e., the diffusing species, making up the surface. R.sub.1 and R.sub.2 are local radii of curvature for points in a small area dA of a curved surface. The radii are perpendicular to the surface and are not parallel if the surface is not planar. When the surface is not planar, the local chemical potential will vary over the surface, resulting in a gradient in the chemical potential that will cause the diffusion of material depending upon external conditions, such as temperature and pressure. It is well known that the flux is proportional to the rate of change of the curvature, i.e., EQU J=(D/kT) .gradient.(.mu..gamma..OMEGA.)=(D.nu..OMEGA./kT).gradient..mu.=(D.gamma..n u..OMEGA..sup.2 /kT) .gradient..sub.s ((1/R.sub.1 (s))+(1/R.sub.2 (s))),
where J is the two-dimensional flux, .nu. is the surface concentration of the diffusing species, D is the coefficient of surface diffusion, k is Boltzmann's constant, T is the temperature, and .gradient..sub.s indicates the gradient with respect to distance along the curved surface. R.sub.1 (s) and R.sub.2 (s) are functions describing the radii of curvature R.sub.1 and R.sub.2, respectively, in orthogonal directions as illustrated in FIG. 12. In other words, along the edge 7 of the unit surface shown in FIG. 12, R.sub.2 is the radius of curvature, and along the edge 8, which is generally orthogonal to the edge 7, the radius of curvature is R.sub.1.
FIG. 11 illustrates the calculation of the movement of a point P.sub.io using the Leon method. The respective radii of curvature at points P.sub.io-1, P.sub.io, P.sub.io+1 respectively are R.sub.io-1, R.sub.io, and R.sub.io+1 so that the flux from point P.sub.io-1 to P.sub.io is given by EQU J.sub.i-1, i =(-D.sub.o /kT) ((1/R.sub.i-1) --(1/R.sub.i))/.vertline.P.sub.1 -P.sub.i-1 .vertline.
where D.sub.o =D.gamma..nu..OMEGA..sup.2.
In FIG. 11, the point P.sub.io advances to the point P.sub.i during a time increment .increment.t. This movement is based upon the difference in the fluxes at the point P.sub.io with respect to points P.sub.i-1 and P.sub.i+l.
In order to apply these calculations in the Leon method, the radii of curvature must be calculated after the surface of interest is divided into various lengths, for example, with respect to FIG. 9, after the two-dimensional figure is divided by the grid points 5. Although this technique may be satisfactory for a two-dimensional application such as that shown in FIG. 9 and as described in the article by Leon and for certain special limited three-dimensional situations, in general, it is extremely difficult to apply the Leon method to generalized three-dimensional surfaces. However, as the density of integration of semiconductor devices has increased, particularly in so-called ultra large scale integration (ULSI), it has become increasingly important to predict reflow surface topographies in three dimensions.